Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{x^2 - 16}{x - 4}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = x$ $ b = \sqrt{16} = -4$ So we can rewrite the expression as: $p = \dfrac{({x} {-4})({x} + {4})} {x - 4} $ We can divide the numerator and denominator by $(x - 4)$ on condition that $x \neq 4$ Therefore $p = x + 4; x \neq 4$